Definition:Jordan Arc
Definition
Let $f: \closedint 0 1 \to \R^2$ be an injective path in the Euclidean plane.
Then $f$ is called a Jordan arc.
Also known as
Some texts refer to a Jordan arc as merely an arc.
Also defined as
Some texts define a Jordan arc $f: \closedint 0 1 \to X$ as an injective path, where $X$ is alternatively defined as:
- the complex plane $\C$
- a real Euclidean space $\R^n$
- a $T_2$ (Hausdorff) topological space $\struct{ S, \tau_S }$
This is what $\mathsf{Pr} \infty \mathsf{fWiki}$ defines as an arc.
Some texts, especially those on complex analysis, drop the condition about injectivity and instead state that:
- $\map f {t_1} \ne \map f {t_2}$ for all $t_1 ,t_2 \in \hointr 0 1$ with $t_1 \ne t_2$
- $\map f t \ne \map f 1$ for all $t \in \openint 0 1$
That is, either $f$ is injective and a Jordan Arc, or $\map f 0 = \map f 1$, when $f$ is a Jordan curve by the $\mathsf{Pr} \infty \mathsf{fWiki}$ definition.
Some texts, especially those on topology, define a Jordan arc as topological subspace $\struct{C, \tau_C}$ of $\R^2$ or $X$, where $\struct{C, \tau_C}$ is homeomorphic to the closed interval $\closedint 0 1$.
This means they consider a Jordan arc to be a topological space rather than a mapping.
Also see
Source of Name
This entry was named for Marie Ennemond Camille Jordan.
Sources
- 2017: Thierry Vialar: Handbook of Mathematics: $10$: Topology: $\S 14$: Theory of Curves